How Do Finance Specialists Think? Talk prepared for the LSE Series, “How Social Scientists Think” Ronald Anderson London School of Economics and CEPR 17th January 2008 1
Outline • The problem of finance • The emergence of modern finance • Finance in a world of market imperfections • Conclusions 2
The problem of finance • Historically, finance has emerged from thinking of economists who have studied the problem of allocating resources over time . Here the key questions are: 1. How much should be saved and invested in projects that will pay off in the future? 2. Among the alternatives available what is the best form for the investments to take? • However, these questions inevitably hits up against the uncertainty of future events. The proper formu- lation of thinking about these problems awaited tools for the study of choice making under uncertainty. 3
Major developments in tools for the analysis of uncertainty • Early building blocks were (a) expected utility theory by von Neumann, Morgenstern, and others and (b) the state contingent claims approach made the key insight that goods we consume can be distinguished by both the date when we consume and the state of nature prevailing when we consume them. • Thus the scope of finance is to study the problem of allocating resources over time and over states of nature. This risk shifting is accomplished in financial contracts and securities. • Arrow, Debreu and others showed that in general equilibrium with complete market securities markets will be efficient in the sense of Pareto. • When some markets are missing Arrow also showed that if securities market allow all risks to be hedged, then markets will be effectively complete . 4
Early applications to finance • Modern portfolio theory by Tobin and Markowitz in the early 1950’s. Allocate wealth based on expected returns on projects and on the variances and covari- ances of all returns. This reaps the benefits of diver- sification . • The Capital Asset Pricing Model (CAPM) was devel- oped by Sharpe, Lintner, Mossin and Treyner about 1960. This was most prominent early example of gen- eral equilibrium analysis to be put to a very practical purpose. The CAPM is an example of a linear pric- ing model : the risk premium should be a linear func- tion of that security’s systematic risk (reflecting the correlation of that market with an index of returns for the market overall). 5
The emergence of modern finance • Diversification of risks , mean-variance analysis , and competitive equilibrium pricing are part of the gen- eral intellectual bagage of all economists these days. What is special about finance? • It think that the key developments that give a distinct finance approach to problems were contributions be- teen the late 1950’s and the late 1970’s building on the economics of uncertainty which made us aware of the power of the logic of arbitrage . • The time period here with reference to the publication of the paper by Modigliani and Miller in 1958 and to the publication in 1979 of the paper by Harrison and Kreps. • Arbitrage is an old notion in economics and underlies the venerable law of one price . However, starting with Modigliani and Miller it was shown that often you can go very far in understanding a problem with- out postulating all the structure about preferences, production technology and so forth as typically re- quired in a fully formulated economic model. 6
What is an arbitrage? • It is the purchase of one collection of goods or secu- rities and the simultaneous sale of another collec- tion of goods and securities which produces a gain in at least one state of nature without incurring a loss in any state of nature. • An arbitrage is a private money machine. As such, people can be relied upon to pursue arbitrage oppor- tunities without limit. • Therefore, prices will be forced to adjust to reflect no arbitrage conditions . Thus the logic of arbitrage leads us to rules for relative pricing of securities . 7
Modigliani-Miller • In the absence of market frictions the value of the equity, debt and other liabilities of the firm must have a total value equal to the value of the assets of the firm. • Merton Miller’s pizza pie analogy: It does not matter whether you cut the pie into many slices or few, it is still has the same total number of calories. • In symbols for a firm with only Debt and Equity, A = D + E • This has many implications. For example, ROE − r = A E ( ROA − r ) where r is return on debt. Higher returns to share- holders come from either increasing returns on assets or increasing leverage (and risk). 8
Modigliani-Miller with state contingent claims • This leads us to an insight as to how values of secu- rities can be related to their expected payoffs in the future. • Suppose that there is a complete market for securities that payoff one unit of numraire in a given state of nature s and nothing otherwise, and let that security’s price be given as π s . • What is the value of a security that pays off 1 unit in every state? This security is equivalent to a portfolio consisting of one unit of each state contingent claims. It is also equivalent to investing 1 / (1 + r ) today at the risk-free rate r . So its no-arbitrage value is, 1 V 0 = s π s = � (1 + r ) • Now let us use the prices π s and the interest rate r to define a set of variables π ∗ s = π s (1+ r ). These must be positive because otherwise there is an arbitrage. By 9
� π ∗ construction they sum to unity s = 1. So this set of modified prices defines a probability distribution over future states of the world. This distribution is called the “risk-neutral” probability distribution in finance. • Apply this to a general security paying a variable amount y s in each state s . Its no arbitrage value is is, π ∗ 1 s (1 + r ) E ∗ y s V = s π s y s = (1 + r ) y s = � � s That is, the value of any security can be expressed as the present discounted value of its expected payoff tomorrow where expectations are taken with respect to the risk neutral probability distribution and dis- counting is done at the risk-free rate. • We stress that if the prices π s are the market prices of the elementary state claims then this expression gives the fair value in the market of the security. Even though agents may be averse to risk, securities are priced in the market as though agents were risk-neutral but calculated expectations using the probability dis- tribution π ∗ s . 10
• Now the market value of the asset V computed in this way should be distinguished from the actuarially fair value of the asset, V a which would be based on the statistical probability distribution over states, π a s , 1 1 V a = s π a s y s = (1 + r ) Ey s � (1 + r ) Note that in Ey s expectation is taken with respect to the statistical distribution of states. • The difference V a − V reflects the discount that the market imposes in order the bear the risk involved in holding the payoffs { y s } . Another way in finance the market value of the security is expressed is using the risk adjusted rate of return on the security r ∗ defined 1 by, V = (1+ r ∗ ) Ey s . 11
Black-Scholes pricing • The famous the Black-Scholes formula for the pric- ing of call options is an extension of these valuation formulae. • Starting with Louis Bachelier (in 1900) and subse- quently by others who found various expressions for the actuarially fair value which in our notation can be expressed as, 1 C a = (1 + r ) EMax ( S T − X, 0) This has two problems: it is not the market value and it involves predicting equity prices, i.e, taking a view on the future course of the stock market. • For example, Paul Samuelson solved for the actuar- ial value under the assumption that the stock price followed a geometric Brownian motion which can be denoted, dS = µSdt + σSdz (1) 12
where dS is the change of the stock price over a very small time interval dt and dz is a Brownian motion, i.e., random variable following a normal distribution over dt . But this solution involves the drift of the stock process µ . • Black and Scholes found that they could construct a dynamic sequence of arbitrage portfolios involv- ing the underlying stock and short term lending such that the portfolio was riskless over short time peri- ods dt . Applying the logic of arbitrage they were able to derive a partial differential equation that must be obeyed by the price of the call option in the absence of arbitrage. They were able to solve that equation and found an expression which can be written in our notation as, 1 (1 + r ) E ∗ Max ( S T − X, 0) C = where the expectation E ∗ is taken with respect to the risk-neutralized process, dS = rSdt + σSdz (2) 13
Let us write this solution as C ( S t , X, r, T, σ ). This is remarkable because it gives us an expression for the fair market value of the option and because its calculation does not require us to take a view on the direction of the stock market, i.e., the parameter µ . 14
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